Break the race down into stages.
It’s important to practice the start. A good start can’t win you a race, but a bad start can certainly lose you one. Practice clipping in quickly, getting off the line and accelerating quickly.
The dismount is probably the most important skill in cyclo-cross and it’s used in order to jump over manmade obstacles – often planks laid out across the course – or to get off the bike on unrideable sections.
The dismount is a fairly specific skill but most people normally pick it up quickly. If you’re coming into an obstacle, have your hands on the hoods and cover the brakes. Change down through the gears so you’re in the correct gear for when you need to remount and pull away again.
Slow down to running speed, unclip your right foot – unless you’re left-handed, which is a different story – swing that round the back of the bike so you’re coasting on one pedal, put your right hand on the toptube as you’re coasting along, then in one swift movement unclip your left foot, hop onto the floor and begin running. With your right hand on the toptube, and the other on the handlebar, you can pick the bike up and either shoulder it or lift it over the obstacle.
As for the remount, put your hands back on the bars so they’re in a good, stable position, then it’s a case of hopping/sliding back on to the saddle and hitting the pedals as hard as you can to get back up to speed while getting clipped in again.
Practice makes perfect. I remember as youth rider having a couple of obstacles setup in the garden and just going round and round and round, practising for hours. Even now during my skills session I practice getting on and off the bike a lot. That’s another great thing about cross. No matter how good you get at it, you can always refine your technique to get better, quicker and smoother.